Research on association schemes and related topics

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The head researcher has done the research in the following 3 directions.1.Study of spin models. (i) We introduced the concept of 4-weight spin models, and proved that link invariants are obtained from them (joint work with Etsuko Bannai). (ii) We gave the complete classifications of the dualities and the modular invariances on the character tables of finite abelian groups, and proved that spin models are constructed from such solutions (joint work with Etsuko Bannai and F.Jaeger).2.Construction of various automorphic forms from either the weight enumerators of codes or polynomial invariants of certain finite groups.(i) We proved that Jacobi forms are constructed from the simultaneous diagonal actions of the 2-dimensional unitary reflection group of order 192 (No.9 in Shepherd-Todd's list)(joint work with Michio Ozeki).(ii) Explicit constructions of certain Jacobi forms of weight 4 (joint work with Michio Ozeki and Shinri Minashima).(iii) We determined the explicit basis of the polynomial invariants mentioned in (i) above (joint work with Etsuko Bannal, Michio Ozeki and Yasuo Teranishi).(iv) We are currently studying Type II additive codes on finite abelian groups (joint work with Masaaki Harada, S.Dougherty, and Manabu Oura).3.I classified primitive symmetric association schemes with m_1=3 by using elementary geometric ideas such as the classifications of regular polyhedrons and quasi-regular polyhedrons. I am currently working on the classification of primitive symmetric Q-polynomial association schemes with m_1=4 jointly with Attila Sali.